Abstract
In the Gaia era, understanding the effects of the perturbations of the Galactic disc is of major importance in the context of dynamical modelling. In this theoretical paper we extend previous work in which, making use of the epicyclic approximation, the linearized Boltzmann equation had been used to explicitly compute, away from resonances, the perturbed distribution function of a Galactic thin-disc population in the presence of a non-axisymmetric perturbation of constant amplitude. Here we improve this theoretical framework in two distinct ways in the new code that we present. First, we use better estimates for the action-angle variables away from quasi-circular orbits, computed from the AGAMA software, and we present an efficient routine to numerically re-express any perturbing potential in these coordinates with a typical accuracy at the per cent level. The use of more accurate action estimates allows us to identify resonances such as the outer 1:1 bar resonance at higher azimuthal velocities than the outer Lindblad resonance, and to extend our previous theoretical results well above the Galactic plane, where we explicitly show how they differ from the epicyclic approximation. In particular, the displacement of resonances in velocity space as a function of height can in principle constrain the 3D structure of the Galactic potential. Second, we allow the perturbation to be time dependent, thereby allowing us to model the effect of transient spiral arms or a growing bar. The theoretical framework and tools presented here will be useful for a thorough analytical dynamical modelling of the complex velocity distribution of disc stars as measured by past and upcoming Gaia data releases.
Highlights
The natural canonical coordinate system of phase-space for Galactic dynamics and perturbation theory is the system of action-angle variables (Binney & Tremaine 2008)
Enough, the linear deformation due to the bar is generally stronger in the Actionbased Galaxy Modelling Architecture (AGAMA) case, and that due to the spiral is weaker in the AGAMA case
This means that reproducing the effect of spiral arms on the local velocity distribution might require a higher amplitude when considering an accurate estimate of the action-angle variables rather than the epicyclic approximation
Summary
The natural canonical coordinate system of phase-space for Galactic dynamics and perturbation theory is the system of action-angle variables (Binney & Tremaine 2008). We use a better estimate than the epicyclic approximation for the action-angle variables, relying on the torus mapping method of McGill & Binney (1990) to convert from actions and angles to positions and velocities, and on the Stäckel fudge (Binney 2012; Sanders & Binney 2016) for the reverse transformation This will allow us to extend previous results to eccentric orbits and orbits wandering well above the Galactic plane. The general idea of torus mapping is to first express the Hamiltonian in the action-angle coordinates (JT , θT ) of a toy potential, for which the transformation to positions and velocities is fully known analytically. In the following we make use of both types of transformations, namely the torus mapping to express the potential in action-angle coordinates and the Stäckel fudge to represent distribution functions in velocity space at a given configuration space point. The equations of motion connecting actions J and the canonically conjugate angles θ are
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